If \(\mathcal {T}\)  is replaced with the class of functions \(a\colon {\mathbb {R}}_+\rightarrow {\mathbb {R}}\)  that satisfy the slightly more restrictive growth condition \(t^{k+\varepsilon }\prec a(t)\prec t^{k+1}\)  for some \(k\in {\mathbb {Z}}_+\) , then the argument used to prove [1]} can be applied without any change to prove Proposition REF . For the more extended class of functions used above one can employ the argument used to prove [2]} without essential changes. We omit the details.
